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Definition of One-to-One Functions - Problem 2
Determining if a set of points is a function and if a set of points is one-to-one. So behind me I have a set of points. Function f is the set of points and I want to determine if this is a function. So to determine whether it's function, for every x there’s only one y. Let’s go through that and what we have is in ascending order our values -1, 0, 1 and 2. There is no repeated x values so f for every x there is only one y, this has to be a function.
Is this one-to-one? So we already have the function part for every x there’s only one y. The other part we are concerned with is, for every y is there only one x? So going through this, the first thing I see is the last two points; where we have the same y value going to two different x values. So here we have the y value 2 corresponding to the x value 1 and 2.
So that tells me this is not a function, for one y value we have two x’s. So what I actually want to do is just change this up to make it a one-to-one function. So I know that this point is a parabola. So what that means is I have to change this 2 value, to something that is not represented in our y’s.
So we have 1 0, 1 2, if I change this to 3, that y value is not represented, so that means for every y 3 there is only that 1 x value of 2. Checking to see if we have any other over lap. So we have our y value 3, 2, 1, 0, 1, there’s also some overlap in these two points.
The y value of 1 corresponds to -2 as x and also 0. So we need to change one of those to a value that’s not represented as well. If we change it to say one of them, to a negative, then we have two different y values with no x value overlap.
So as this function was not one-to-one but with some minor tweaks, which you're probably not going to be able to do in normal application just changing the values. But we are also able to make this into what could be a one-to-function.